5L 3s

5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of 5edo = 480¢) to 3\8 (three degrees of 8edo = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this:

generator					tetrachord	g in cents	2g	3g	4g	Comments
2\5					1 0 1	480.000	960.000	240.00	720.000
21\53					10 1 10	475.472	950.943	226.415	701.887	Vulture/Buzzard is around here
19\48					9 1 9	475	950	225	700
17\43					8 1 8	474.419	948.837	223.256	697.674
15\38					7 1 7	473.684	947.368	221.053	694.737
13\33					6 1 6	472.727	945.455	218.181	690.909
11\28					5 1 5	471.429	942.857	214.286	685.714
9\23					4 1 4	469.565	939.130	208.696	678.261	L/s = 4
					pi 1 pi	467.171	934.3425	201.514	668.685	L/s = pi
7\18					3 1 3	466.667	933.333	200.000	666.667	L/s = 3
					e 1 e	465.535	931.069	196.604	662.139	L/s = e
	19\49				8 3 8	465.306	930.612	195.918	661.2245
		50\129			21 8 21	465.116	930.233	195.349	660.465
			131\338		55 21 55	465.089	930.1775	195.266	660.335
				212\547	89 34 89	465.082	930.1645	195.247	660.329
			81\209		34 13 34	465.072	930.1435	195.215	660.287
		31\80			13 5 13	465	930	195	660
	12\31				5 2 5	464.516	929.032	193.549	658.065
5\13					2 1 2	461.538	923.077	184.615	646.154
					√3 1 √3	459.417	918.8345	178.252	637.669
	13\34				5 3 5	458.824	917.647	176.471	635.294
		34\89			13 8 13	458.427	916.854	175.281	633.708
			89\233		34 21 34	458.369	916.738	175.107	633.473
				233\610	89 55 89	458.361	916.721	175.082	633.443	Golden father
			144\377		55 34 55	458.355	916.711	175.066	633.422
		55\144			21 13 21	458.333	916.666	175	633.333
	21\55				8 5 8	458.182	916.364	174.545	632.727
					pi 2 pi	457.883	915.777	173.665	631.553
8\21					3 2 3	457.143	914.286	171.429	628.571	Optimum rank range (L/s=3/2) father
11\29					4 3 4	455.172	910.345	165.517	620.690
14\37					5 4 5	454.054	908.108	162.162	616.216
17\45					6 5 6	453.333	906.667	160	613.333
20\53					7 6 7	452.83	905.66	158.491	611.321
23\61					8 7 8	452.459	904.918	157.377	609.836
26\69					9 8 9	452.174	904.348	156.522	608.696
29\77					10 9 10	451.948	903.896	155.844	607.792
3\8					1 1 1	450.000	900.000	150.000	600.000

The only notable harmonic entropy minimum is Vulture/Buzzard, in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.

By a weird coincidence, the other generator for this MOS will generate the same pattern within a tritave equivalence. By yet another weird coincidence, this MOS belongs to a temperament which has Bohlen-Pierce as its index-2 subtemperament. In addition to being harmonious, this tuning of the MOS gives an L/s ratio between 3/1 and 3/2, which is squarely in the middle of the range, being thus neither too exaggerated nor too equalized to be recognizable as such, unlike in octaves, where the only notable harmonic entropy minimum is near a greatly exaggerated 10/1 L/s ratio.

generator					tetrachord	g in cents	2g	3g	4g	Comments
2\5					1 0 1	760.782	1521.564	380.391	1141.173
27\68					13 1 13	755.188	1510.376	363.609	1118.797
					~6626 515 6626	755.132	1510.265	363.442	1118.574	2g=12/5 minus quarter comma
25\63					12 1 12	754.744	1509.488	362.277	1117.021
23\58					11 1 11	754.2235	1508.447	360.716	1114.939
21\53					10 1 10	753.605	1507.21	358.859	1112.464
19\48					9 1 9	752.857	1505.714	356.617	1109.474
17\43					8 1 8	751.936	1503.871	353.852	1105.788
15\38					7 1 7	750.771	1501.543	350.36	1101.132
	28/71				13 2 13	750.067	1500.1335	348.245	1098.312
	41\104				19 3 19	749.809	1466.618	347.4725	1097.282	3g=11/3 near here
13\33					6 1 6	749.255	1498.51	345.81	1095.065
	24\61				11 2 11	748.31	1496.62	342.976	1091.286
	35\89				16 3 16	747.96	1495.92	341.924	1089.884
	46\117				21 4 21	747.777	1495.554	341.377	1089.154
	57\145				26 5 26	747.665	1495.33	341.04	1088.705
					5+√29 2 5+√29	747.648	1495.297	340.99	1088.638
	68\173				31 6 31	747.589	1495.178	340.813	1088.402
		147\374			67 13 67	747.56	1495.12	340.725	1088.285	4g=45/8 near here
	79\201				36 7 36	747.535	1495.069	340.649	1088.183
11\28					5 1 5	747.197	1494.393	339.635	1086.831
	20\51				9 2 9	745.865	1491.729	335.639	1081.50
	29\74				13 3 13	745.361	1490.721	334.127	1079.488
	38/97				17 4 17	745.096	1490.192	333.332	1078.428
					2+√5 1 2+√5	754.051	1490.101	333.197	1078.247
	47\120				21 5 21	744.932	1489.865	332.842	1077.7745
9\23					4 1 4	744.243	1488.487	330.775	1075.018	L/s = 4
	43\110				19 5 19	743.4915	1486.983	328.5195	1072.011
		77\197			34 9 34	743.404	1486.807	328.256	1071.66	4g=39/7 near here
	34\87				15 4 15	743.293	1486.586	327.923	1071.216
	25\64				11 3 11	742.951	1485.902	326.899	1069.85
	16\41				7 2 7	742.226	1484.453	324.724	1066.95
	23\59				10 3 10	741.44	1482.88	322.365	1063.805
					3+√13 2 3+√13	741.289	1482.577	321.911	1063.20
	30\77				13 4 13	741.021	1482.043	321.109	1062.131
					pi 1 pi	740.449	1480.898	319.392	1056.841	L/s = pi
7\18					3 1 3	739.649	1479.298	316.992	1056.642	L/s = 3
	89\229				38 13 38	739.188	1478.376	315.608	1054.796	3g=18/5 near here
	82\211				35 12 35	739.148	1478.297	315.49	1054.639
	75\193				32 11 32	739.102	1478.203	315.35	1054.452
	68\175				29 10 29	739.045	1478.091	315.181	1054.227
	61/157				26 9 26	738.976	1477.952	314.973	1053.949
	54\139				23 8 23	738.889	1477.778	314.712	1053.601
	47\121				20 7 20	738.776	1477.552	314.373	1053.149
	40\103				17 6 17	738.623	1477.247	313.915	1052.538
	33\85				14 5 14	738.406	1476.812	313.263	1051.669
	26\67				11 4 11	738.072	1476.144	312.261	1050.333
					e 1 e	737.855	1478.71	311.61	1049.465	L/s = e
	19\49				8 3 8	737.493	1474.986	310.523	1048.016
			164\423		69 26 69	737.401	1474.802	310.248	1047.649	3g=18/5 minus quarter comma near here
			145\374		61 23 61	737.389	1474.778	310.212	1047.601
			126\325		53 20 53	737.373	1474.747	310.165	1047.538
			107\276		45 17 45	737.352	1474.704	310.101	1047.453
			88\227		37 14 37	737.322	1474.644	310.01	1047.332
			69\178		29 11 29	737.275	1474.549	309.869	1047.144
		50\129			21 8 21	737.192	1474.384	309.621	1046.812
			131\338		55 21 55	737.148	1474.296	309.49	1046.638
				212\547	89 34 89	737.138	1474.276	309.459	1046.597
			81\209		34 13 34	737.121	1474.243	309.409	1046.53
		31\80			13 5 13	737.008	1474.015	309.068	1046.075
	12\31				5 2 5	736.241	1472.481	306.767	1043.007
					1+√2 1 1+√2	735.542	1471.084	304.6715	1040.214	Silver false father
	17\44				7 3 7	734.846	1469.693	302.584	1037.41
	22\57				9 4 9	734.088	1468.176	300.309	1034.397
	27\70				11 5 11	733.611	1467.222	298.879	1032.49
		59\153			24 11 24	733.434	1466.867	298.346	1031.779
	32\83				13 6 13	733.284	1466.568	297.897	1031.181	2g=7/3 near here
5\13					2 1 2	731.521	1463.042	292.609	1024.13
	53\138				21 11 21	730.461	1460.922	289.428	1018.889
		101\263			40 21 40	730.409	1460.817	289.271	1019.679	3g=39/11 near here
	48\125				19 10 19	730.35	1460.701	289.097	1019.448
	43\112				17 9 17	730.215	1460.43	288.69	1018.905
	38\99				15 8 15	730.043	1460.087	288.175	1018.218
		71\185			28 15 28	729.9395	1459.879	287.8635	1017.803
		104\271			41 22 41	729.902	1459.803	287.75	1017.651	4g=27/5 near here
	33\86				13 7 13	729.82	1459.64	287.505	1017.325
	28\73				11 6 11	729.547	1459.034	286.596	1016.113
	23\60				9 5 9	729.083	1458.1655	285.293	1014.376
		41\107			16 9 16	728.7865	1457.573	284.4045	1013.191	3g=99/28 near here
		59\154			23 13 23	728.671	1457.342	284.058	1012.729
		77\201			30 17 30	728.61	1457.219	283.874	1012.483
		95\248			37 21 37	728.5715	1457.143	283.7595	1012.331	Golden BP is index-2 near here
	18\47				7 4 7	728.408	1456.817	283.27	1011.678
					√3 1 √3	728.159	1456.318	282.522	1010.6815
		49\128			19 11 19	728.092	1456.184	282.321	1010.413	4g=27/5 minus third comma near here
		31\81			12 7 12	727.909	1455.817	281.771	1009.68
	13\34				5 3 5	727.218	1454.436	279.699	1006.917
		34\89			13 8 13	726.59	1453.179	277.814	1004.403
			89\233		34 21 34	726.498	1452.996	277.538	1004.036
				233\610	89 55 89	726.4845	1452.969	277.4985	1003.983	Golden false father
			144\377		55 34 55	726.476	1452.952	277.473	1003.95
		55\144			21 13 21	726.441	1452.882	277.368	1003.809
	21\55				8 5 8	726.201	1452.402	276.468	1002.849
					pi 2 pi	725.736	1451.472	275.252	1000.988
8\21					3 2 3	724.554	1449.109	271.708	996.226	Optimum rank range (L/s=3/2) false father
					~543 361 543	724.511	1449.022	271.579	996.09	4g=16/3
	27\71				10 7 10	723.279	1446.557	267.881	991.16
		46\121			17 12 17	723.057	1446.115	267.217	990.274
		65\171			24 17 24	722.965	1445.931	266.941	989.907	3g=7/2 near here
	19\50				7 5 7	722.743	1445.486	266.274	989.017
11\29					4 3 4	721.431	1442.862	262.338	983.77
	25\66				9 7 9	720.4375	1440.875	259.3575	979.795
		64\169			23 18 23	720.267	1440.534	258.848	979.113
			167\441		60 47 60	720.2415	1440.483	258.7965	979.001
				437\1154	157 123 157	720.238	1440.475	258.758	978.996
			270\713		97 76 97	720.235	1440.471	258.751	978.987
		103\272			37 29 37	720.226	1440.451	258.722	978.947
	39\103				14 11 14	720.158	1440.315	258.518	978.676
14\37					5 4 5	719.659	1439.317	257.021	976.679
	31\82				11 9 11	719.032	1438.064	255.14	974.172
		79\209			28 23 28	718.921	1437.842	254.807	973.728
			206\545		73 60 73	718.904	1437.808	254.757	973.661
				539\1426	191 117 191	718.902	1437.803	254.75	973.652
			333\881		118 97 118	718.90	1437.80	254.745	973.6455
		127\336			45 37 45	718.893	1437.787	254.726	973.619
	48\127				17 14 17	718.849	1437.698	254.592	973.441
17\45					6 5 6	718.516	1437.032	253.549	972.11
20\53					7 6 7	717.719	1435.438	251.202	968.9205
					~401 344 401	717.695	1435.3905	251.131	968.826	4g=21/4
23\61					8 7 8	717.131	1434.261	249.437	966.567
					~6682 5875 6682	716.9925	1433.985	249.0225	966.015	6g=12
26\69					9 8 9	716.679	1433.357	248.081	964.76
29\77					10 9 10	716.321	1432.641	247.007	963.328
32\85					11 10 11	716.03	1432.06	246.135	962.1655
35\93					12 11 12	715.7895	1431.759	245.4135	961.203
38/101					13 12 13	715.587	1431.174	244.806	960.393	2g=16\7 near here
3\8					1 1 1	713.233	1426.466	237.744	950.9775

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